Simultaneous measurability in quantum theory

Abstract

This paper presents a study of what is sometimes regarded as the conceptual heart of quantum theory, namely, the orthodox ‘physical’ interpretation of non-commuting operators as representatives of incompatible (non-simultaneously-measurable) observables. To provide a firm foundation for the analysis, a definite statement of the essentials of modern quantum theory is given briefly in the form of a mathematical axiomatization together with a review of the two measurement constructs introduced elsewhere (Park, 1967b). Contrary to custom in discussions on simultaneous measurability, the uncertainty principle is not dwelt upon but simply stated carefully in order to establish its actual irrelevance to the problem at hand. It is then demonstrated that the much quoted ‘principle’ of incompatibility of noncommuting observables is false. The axiomatic root of all incompatibility arguments is next identified; and it is shown that, with a slight modification of the basic postulates which affects neither useful theorems nor practical calculations, quantum physics no longer entails illogical restrictions on measurability. Among the related topics touched upon are the problem of joint probability distributions, the ‘logical’ approach to quantum mathematics (wherein noncommutativity becomes incompatibility within a propositional calculus), and the field theoretic attempt to unify quantal and relativistic physics through a postulated connection between incompatibility and space-like intervals.